Factor the following expression: $5$ $x^2$ $-24$ $x+$ $27$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(27)} &=& 135 \\ {a} + {b} &=& & & {-24} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $135$ and add them together. The factors that add up to ${-24}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-9}$ and ${b}$ is ${-15}$ $ \begin{eqnarray} {ab} &=& ({-9})({-15}) &=& 135 \\ {a} + {b} &=& {-9} + {-15} &=& -24 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {5}x^2 {-9}x {-15}x +{27} $ Group the terms so that there is a common factor in each group: $ ({5}x^2 {-9}x) + ({-15}x +{27}) $ Factor out the common factors: $ x(5x - 9) - 3(5x - 9) $ Notice how $(5x - 9)$ has become a common factor. Factor this out to find the answer. $(5x - 9)(x - 3)$